Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h _n : 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {V _Ψ (〈c, d〉; f ? h _n )} _n=1 ^∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ? with finite Φ-variation and the functions h _n . In Theorem 2, the same is done for a sequence of functions h _n : ? → ?, n = 1, 2, ..., and the sequence of Ψ-variations {V _Ψ(〈a, b〉; h _n ? f)} _n=1 ^∞ .     

Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation (1) $$- \frac{d}{{dx}}K(x,\frac{{du}}{{dx}}) + f(x,u) = 0,0< x< 1,$$ with boundary conditions (2) $$u(0) = u(1) = 0$$ by difference boundary-value problems of form (3) $$- \left[ {a(x,w_{\bar x} )} \right]_x + \varphi (x,w) = 0,x \in w_r ,$$ (4) $$w(0) = w(1) = 0.$$ Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the functionf(x,u(x)).     

Laurent expansion of an inverse of a function matrix

This paper suggests a general procedure based on the Taylor expansion of a function matrixF(z) for calculating the Laurent expansion ofF ^?1(z) around an isolated pole. It is shown that in order to compute thejth Laurent coefficient matrixB _j ofF ^?1(z), one needs in any case the Taylor coefficientsA ₀, A₁,..., A_2n+j ofF(z), wheren is the order of the pole. Theorem 1 helps to determine the order of the pole, while Theorem 2 shows also how the Laurent coefficients can be computed in the general case.     

Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

Bombieri's theorem in short intervals

The well-known Bombieri-A. I. Vinogradov theorem states that (1) $$\sum\limits_{q \leqslant x^{\tfrac{1}{2}} (\log x)^{ - s} } {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{y \leqslant x} } \left| {\psi (y,q;a) - \frac{y}{{\varphi (q)}}} \right| \ll \frac{x}{{(\log x)^A }},$$ whereA is an arbitrary positive constant,B=B(A)>0, and as usual, $$\psi (x,q;a) = \sum\limits_{\mathop {n \leqslant x}\limits_{n = a(q)} } {\Lambda (n),}$$ Λ being the Von Mangoldt's function. The problem of finding a result analogous to (1) for short intervals was investigated by many authors. Using Heath-Brown's identity and the approximate functional equation for DirichletL-functions, A. Perelli, J. Pintz and S. Salerno in 1985 established the following extension of Bombieri's theorem: Theorem 1. (2) $$\sum\limits_{q \leqslant Q} {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{h \leqslant y} \mathop {\max }\limits_{\frac{x}{2}< \approx \leqslant x} } \left| {\psi (z + h,q;a) - \psi (z,q;a) - \frac{h}{{\varphi (q)}}} \right| \ll \frac{y}{{(\log x)^A }}$$ where A>0 is an arbitrary constant,y=x ^θ $$\frac{7}{{12}}< \theta \leqslant 1, Q = x^{\frac{1}{{40}}} .$$ ,Q=x ^1/40. By improving the basic lemma which A. Perelli, J. Pintz and S. Salerno used as the main tool to prove Theorem 1, we obtain Theorem 2.Under the same condition as in Theorem 1,for Q=x ^1/38.5, (2)still holds.     

Sufficient triangular norms in many-valued logics with standard negation

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms.     

Resolution of composite fuzzy relation equations based on Archimedean triangular norms

Lately, the sup-t-norm composition of fuzzy relations has been used instead of the well-known max–min. Thus, there is a need for methods of studying and solving sup-t-norm fuzzy relation equations (t is any t-norm). In this paper, the solution existence problem is first studied and solvability criteria for composite fuzzy relation equations of any t-norm are given. Then, a methodology for solving fuzzy relation equations based on sup-t composition, where t is an Archimedean t-norm, is proposed. This resolution method is simpler and faster than those proposed for covering all the continuous t-norms. The result is important, since, as is shown in the paper, the only continuous t-norm that is not Archimedean is the “minimum”.     

Polynomials of least deviation from zero in the L[−1, 1] metric with five fixed coefficients

We study properties of polynomials R _n+5(x) of least deviation from zero in the L[?1, 1] metric, with five given leading coefficients whose forms were calculated previously. Theorems 1 and 2 together with Theorem A contain, in particular, a final classification of polynomials R _n+5(x) that have exactly (n + 1) sign changes in (?1, 1).     

The generalized analytic signal

This paper is an explication of the analytic signal in the generalized case, i.e., the analytic signal of a generalized function and of a generalized stochastic process. The contributions of the author are: (1) an L²-theory of distributions which, in the study of the analytic signal, has an advantage over the usual Schwartz-Itô-Gel'fand theory because the Cauchy representation is defined; (2) a proof (Theorem 2.5) that the Schwartz distributions δ, δ⁺, δ^? and ? may be extended to the L₂-case, expressions (Theorems 2.6 and 2.7) for their Hilbert and Fourier transforms in the L₂-case, and expressions (Section 2.1) for their analytic signals; (3) a proof (Theorem 3.3) that an orthogonal L₂-process, and therefore the Fourier transform of a second-order stationary stochastic process (Theorem 3.4), is strictly generalized; (4) a representation theorem (Theorem 3.5) which extends the Itô spectral representation theorem for stationary random distributions to the nonspectral, nonstationary, L₂-case; (5) expressions for the Cauchy representation (Theorem 3.6) and the analytic signal (Theorem 3.7) of an L₂-process; (6) an expression for and the covariance kernel of the analytic signal of white noise (Section 3.4). The word application in the text refers to the application of previously developed concepts.     

Total controllability to affine manifolds of control systems

A system is totallyG-controllable if every pointx ₀ of the state spaceE ⁿ can be steered to the targetG in finite time and can be held inG forever afterward. Sufficient conditions are developed for the totalG-controllability of the linear system (a) $$\dot x(t) = A(t)x(t) + B(t)u(t)$$ and its perturbation (b) $$\dot x(t) = A(t)x(t) + B(t)u(t) + F(t,x(t),u(t)),$$ where the targetG is an affine manifold inE ⁿ. We state conditions on the perturbation functionF which guarantee that, if (a) is totallyG-controllable, then so is (b). These conditions onF are natural and are obtained by solving a system of nonlinear integral equations by the Leray-Schauder fixed-point theorem.     

Improved error bounds for near-minimax approximations

Givenf εC ⁽ⁿ⁺¹⁾[?1, 1], a polynomialp _n, of degree ≤n, is said to be near-minimax if (*) $$\left\| {f - p_n } \right\|_\infty = 2^{ - n} |f^{(n + 1)} (\xi )|/(n + 1)!,$$ for some ζ ε (?1,1). For three sets of near-minimax approximations, by considering the form of the error ∥f ?p _n∥_∞ in terms of divided differences, it is shown that better upper and lower bounds can be found than those given by (*).     

The omni-transform in the renewal model and in single-channel queues

Some basic results of the renewal model are effectively summarized by (1) $$E\psi '(r) = E[\psi (x) - \psi (0)]/Ex,$$ wherex is the random variableservice time, r is its associatedresidual time, Ψ ( ) is an arbitrary “well behaved” function, andE is the expectation operator. The process “waiting time for service of a new arrival”, denoted byw, is effectively summarized in the modelM/G/1 by (2) $$E\psi (w) = (1 - \rho )\psi (0) + \rho E\psi (w + r).$$ We refer toEΨ (Z) as theomni-transform of the random variable or processZ, and to equations typified by (2) asomni-equations, i.e. equations valid for an arbitrary well-behaved functionΨ ( ). The omni-transform owes its flexibility to the arbitrariness ofΨ ( ) and its ease of handling to its simplicity when applied to mixtures and sums of random variables. From (2) we obtain the moments ofw by puttingΨ (w)=w ^k , the Laplace transform ofw by puttingΨ(w)=e ^?sw , and the convolution equation (2a) for the distribution ofw by puttingΨ(w)=1 ifw≥t andΨ(w)=0 otherwise: (2a) $$\Pr (w \leqslant t) = (1 - \rho ) + \rho \Pr (w + r \leqslant t),$$ a result equivalent to the Takacs integro-differential equation. Using repeatedly the so-called shift property of omni-equations, (2a) can be solved by representing the distribution ofw as an infinite series of convolutions: (3) $$\Pr (w \leqslant t) = (1 - \rho ) + (1 - \rho )\rho \Pr (r_1 \leqslant t) + (1 - \rho )\rho ^2 \Pr (r_1 + r_2 \leqslant t) + + ,$$ where ther _i are a set of independent random variables, each distributed liker. Equation (3) is equivalent to a theorem by Benes. An analogy between the process “waiting time inM/G/1” and the process “toss a coin till heads shows up” where the tossing time is a random variable is also pointed out. The omni-calculus also sheds some light on the modelG/G/1. In forthcoming publications, we will apply the omni-calculus to the process “number in queue” inM/G/1, to the analysis of the busy period inM/G/1, and to some modifiedM/G/1 models, e.g. a vacationing server. In these publications too, the omni-method lifts the “Laplace veil” from much of the physical reality underlying the models considered.     

A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

Regular left-continuous t-norms

A left-continuous (l.-c.) t-norm ⊙ is called regular if there is an n<ω such that the map x ↦ x⊙a has, for any a∈[0,1], at most n discontinuity points, and if the function mapping every a∈[0,1] to the set behaves in a specifically simple way. The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras. With each regular l.-c. t-norm, we associate certain characteristic data, which in particular specifies a finite number of constituents, each of which belongs to one out of six different types. The characteristic data determines the t-norm to a high extent; we focus on those t-norms which are actually completely determined by it. Most of the commonly known l.-c. t-norms are included in the discussion. Our main tool of analysis is the translation semigroup of the totally ordered monoid ([0,1];≤,⊙,0,1), which consists of commuting functions from the real unit interval to itself.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Asymptotic behavior of number fields with prescribed ℓ-class numbers

Let K be a cyclic Galois extension of the rational numbers Q of degree ?, where ? is a prime number. Let h_? denote the order of the Sylow ?-subgroup of the ideal class group of K. If h_? = ?^s(s ≥ 0), it is known that the number of (finite) primes that ramify in K/Q is at most s + 1 (or s + 2 if K is real quadratic). This paper shows that “most” of these fields K with h_? = ?^s have exactly s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic). Furthermore the Sylow ?-subgroup of the ideal class group is elementary abelian when h_? = ?^s and there are s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic).     

Zur Bestimmung vonE n [x n+2m ]

Remark on the estimation ofE _n [x ^n+2m ]. Let be $$E_n [f]: = \mathop {\inf }\limits_{p \in P_n } \mathop {\sup }\limits_{x \in [ - 1, 1]} |f(x) - p(x)|$$ (P _n : set of all polynomials of degreen). Riess-Johnson [4] proved (3) $$E_n [x^{n + 2m} ] = \frac{{n^{m - 1} }}{{2^{n + 2m - 1} (m - 1)!}}[1 + O(n^{ - 1} )],n even.$$ This degree of approximation is realized by expansion in Chebyshev polynomials and by interpolation at Chebyshev nodes. The purpose of this paper is to give a more precise estimation by constructing the polynomial of best approximation on a finite set. This construction is easily done and one obtains the result, that the termO(n ^?1) in (3) may be replaced by 1/2(m ? 1) (3m + 2)n ^?1 + O(n ^?2).     

Взаимоотношение меж ду сходимостью подпоследовательно стей частных сумм числово го ряда и его суммируе мостью методами (C, α) и Абеля

Numerical series \(\mathop \Sigma \limits_{n = 0}^\infty u_n\) with partial sumss _n are studied under the assumption that a subsequence \(\left\{ {S_{n_k } } \right\}_{k = 0}^\infty\) of the partial sums is convergent. Then a sequence {η _k } is chosen, by means of which a majorant of the termsu _n is constructed. Conditions on {n _k } and {η _k } are found which imply the (C, 1)-summability of the series∑ u _n (Theorem 1). In the meanwhile, it is proved that the (C, 1)-means in Theorem 1 cannot be replaced by (C, α)-means, if 0<α<1 (Theorem 2). On the other hand, if the assumption in Theorem 1 is not satisfied, then in certain cases the series∑ u _n preserves the property of (C, 1)-summability (Theorems 4 and 5), while in other cases it is not summable even by Abel means (Theorems 3 and 6).     

Moment inequalities and the Riemann hypothesis

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫ ₀ ^∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments \(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turán inequalities (*) $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ We give here a constructive proof that log \(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.